Talking mathematics
Mathematicians are far more talkative than you might think
But often it’s impossible to understand what they are saying to
each other. Leading mathematician Jan Bouwe van den Berg
succeeds in giving the layman interesting insights into his
discipline.“The important thing is to ask the right questions.”
By Rianne Lindhout
Shortly before our interview, mathematician Jan Bouwe van den
Berg received a week-long visit from a postdoc from the US. They
discussed equations they were unable to solve by themselves, then
continued to work on them individually, incorporating the new input
from their discussions. No high-tech equipment, just good old pen
and paper. When one of them had an idea, he tried it out on the
blackboard in Van den Berg’s office. The other offered contributions
along the way and using this method, they were able to make some
progress.
Van den Berg explains: “We only used the computer if there was too
much ... How can I put this clearly? ... If we had to verify whether a
large number of things were true. If we had to solve a great many
equations along the lines of: is the outcome negative or not?”
There you have it: maths is hard to explain. Thanks to the work of
Jan Bouwe van den Berg, Professor of Differential Equations and
Applications at VU University Amsterdam, aircraft will hopefully be
able to run on less fuel in the future. If his line of research does not
turn out to be a dead end, in five to ten years’ time his calculations
will improve our understanding of gas and liquid flows.
Understanding how air flows around aircraft, for example, can help
aerospace designers develop wings with greater lift. “The complex
air flow at the edge of an aircraft’s wing has a particularly strong
influence on the lift,” Van den Berg reveals. That’s the kind of flow
he is trying to fathom using mathematics.
Yet Van den Berg never thinks in terms of airflow when he arrives
for work at the university in the morning. Like all mathematicians,
he has a deep appreciation of the beauty of mathematics.
Appreciating the beauty of mathematics
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it started with an idea
“It’s a real kick to go from not being
able to understand a problem at first
and, by thinking and talking about it, to
finally reach the point where you do
understand”
Jan Bouwe van den Berg: “It’s a real kick to go from not being able
to understand a problem at first and, by thinking and talking about
it, to finally reach the point where you do understand.”] After
graduating from high school, he opted to study mathematics and
physics. “In the sciences, things fall into place. It’s not the same with
languages. There are never exceptions to the rules.” Now
mathematics is his full-time job. “It’s a real kick to go from not
being able to understand a problem at first and, by thinking and
talking about it, to finally reach the point where you do understand.”
Smart people only
He is exceptionally good at his job. Five years ago, at the tender age
of 34, he became a professor. This year, he was awarded 1.5 million
euros by the Netherlands Organization for Scientific Research (NWO)
to recruit talented people for his research programme. This grant is
a remarkable achievement, as nowadays the funding set aside for
the exact sciences is usually snapped up by the computer scientists
and astronomers.
So what exactly is it that makes Van den Berg so good? He laughs:
“In maths there are very smart people only! That’s the basis on which
they are selected.” Then, after some thought: “The important thing is
to ask the right questions. You can ask questions that are too obvious
and therefore not interesting, but it’s also very easy to ask questions
that are far too tough. Questions you’ll never be able to answer.
Apparently asking the right questions is something I know how to
do.”
To have a career in science, those in a position of power have to see
you as a suitable candidate to occupy a certain position. You have to
be able to organize things and supervise other researchers. “People
from the exact sciences tend to be less talented in these areas than
those from the more social disciplines,” acknowledges Van den
Berg. From their earliest beginnings right up to the early stages of
their PhD research, mathematicians are hardly ever selected on
communication skills. After that stage you start teaching, passing on
your knowledge. In other words, being able to deal with other people
suddenly becomes important. For Van den Berg that’s not an issue:
“It’s wonderful to observe how a student who doesn’t understand
something at first suddenly realizes how it all fits together.” He also
regards supervising PhD students as one of the most rewarding
aspects of his work. “I talk to them on a weekly basis and we discuss
what they have done, what they have understood.”
Good luck
As Van den Berg sees it, you won’t go far in mathematics simply by
thinking ‘I’m very smart, so I’m bound to make it’. “There are an
awful lot of very smart people around. For most it’s a matter of
working very hard in order to go far.” This applies to him too, but he
has also had his share of good luck. “I’ve ended up in a field where
there are opportunities for growth, because of the possible
applications. As a PhD student, it’s difficult to see where the
prospects lie. It helps if the right people cross your path.”
In addition, Van den Berg wisely pursued a different branch of
mathematics after obtaining his doctorate, so that he would not be
limited to a single area later on. “That’s an approach I’d recommend
to every mathematician. It’s tricky though: as a postdoc you are
supposed to start publishing your work quickly and that’s hard when
you still have to get up to speed in a new area of mathematics.”
Building block
One of the discoveries that the VU’s Professor of Differential
Equations made involves bringing a pan of liquid to the boil. “We had
a certain type of equation to describe fluids. If you heat fluids from
below, it gives rise to rotating flows called convection rolls. You can
summarize these in a differential equation. Working with a colleague,
I have found a new way to look at this equation, namely by looking at
the solutions as knotted solutions.”
“It’s like contributing a building block to
the edifice of mathematics”
The language of mathematics
The purpose of the mathematics on which Van den Berg usually
works – differential equations – is not to solve equations. He tries to
determine whether the solution is very large or very small. Or even
whether it runs in circles. The equations on which he works concern
things that change over time, for example liquids that flow, that go
from quiet to turbulent, and vice versa. “I’m looking for ways to
describe these transitions.” In mathematical terms, a gas is also a
liquid. The way in which air swirls around an aircraft therefore falls
under the same branch of mathematics used to describe the abovementioned flows in a liquid that is heating up or how a hazardous
substance spreads as it leaks into water. “The language you can use
to describe these phenomena is mathematics.”
doubt
Talking to Professor Van den Berg, you come to understand – to
some extent at least – those satisfying eureka experiences you can
have as a mathematician. But they don’t tell the whole story. Van
den Berg believes that mathematicians may well be more prone to
serious doubts than many scientists. “As a student, you attend
lectures by people who understand everything. You think, this
seems logical, but how do you come up with it in the first place?” As
a PhD student, things only get more challenging. “You have to
design your own research, working among people with at least ten
years’ experience. You only have yourself to rely on. For every one
year in which you feel you’re getting somewhere, three years can go
by in which nothing happens. At conferences, people present years
of research in fifteen minutes. That’s a humbling experience.
Eventually you learn to put these things into perspective, but still.
This isn’t something PhD students share with each other. They tend
only to talk about their successes. I encourage them to share the
ups and the downs.”
“For every one year in which you feel
you’re getting somewhere, three years
can go by in which nothing happens”
Hmm ... “That means that these convection rolls can occur in so
many different ways that you can never predict exactly how they will
move or how many there will be. In a nutshell, the question was: is
the occurrence of such rolls chaotic or not? The answer is yes.” You
could also have demonstrated this empirically, Van den Berg adds.
But his discovery marked a new way of looking at the differential
equation, a viewpoint that can generate yet more innovations. “It’s
like contributing a building block to the edifice of mathematics. And
larger sections of that edifice can be applied in the world of
technology.”
Van den Berg also encounters such feelings of insignificance
himself. He discusses them with his colleagues and creates a bit of
breathing space by tackling another mathematical problem every
now and then. During the annual mathematics study group with
industry, for example, when about fifty mathematicians, including
many PhD students, work together in small groups for a week to
tackle problems presented by companies. Last February, Van den
Berg worked on a question asked by Philips about how to make LED
light softer and less white. “I know a lot about various bits of
mathematics. The trick is to know which bit you should use to
address a particular question. Philips can now build on the good
ideas we came up with.”
it started with an idea
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